Classroom Resource

What Are Perfect Squares?: Algebra 1, Episode 19: Unit 7, Lesson 11 | Illustrative Math

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This video lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of perfect squares. They apply the distributive property repeatedly to expand perfect-square expressions given in the factored form (MP8). The repeated reasoning allows them to generalize expressions of the form (x + n)2 as equivalent to x2 + 2nx + n2.

The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.

Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.5

Use the structure of an expression to identify ways to rewrite it.

Unpacked Content

UP:MA19.A1.5

Vocabulary

  • Terms
  • Linear expressions
  • Equivalent expressions
  • Difference of two squares
  • Factor
  • Difference of squares

Knowledge

Students know:
  • Algebraic properties.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression.

Skills

Students are able to:
  • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.

Understanding

Students understand that:
  • Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.6

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Unpacked Content

UP:MA19.A1.6

Vocabulary

  • Quadratic expression
  • Zeros
  • Complete the square
  • Roots
  • Zeros
  • Solutions
  • x-intercepts
  • Maximum value
  • Minimum value
  • Factor
  • Roots
  • Exponents
  • Equivalent form
  • Vertex form of a quadratic expression

Knowledge

Students know:
  • Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Skills

Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Factor quadratic expressions.
  • Complete the square in quadratic expressions.
  • Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

Understanding

Students understand that:
  • Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.9

Select an appropriate method to solve a quadratic equation in one variable.

Unpacked Content

UP:MA19.A1.9

Vocabulary

  • Completing the square
  • Quadratic equations
  • Quadratic formula
  • Inspection
  • Imaginary numbers
  • Binomials
  • Trinomials

Knowledge

Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • Notational methods for expressing complex numbers.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

Skills

Students are able to:
  • Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
  • Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
  • Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
  • Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
  • Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Understanding

Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.11

Solve quadratic equations with real coefficients that have complex solutions.

Unpacked Content

UP:MA19.A2.11

Vocabulary

  • Complex solution
  • Quadratic equation
  • Real coefficients

Knowledge

Students know:
  • strategies for solving quadratic equations

Skills

Students are able to:
  • apply the quadratic equation.
  • provide solutions in complex form.

Understanding

Students understand that:
  • all quadratic equations have two solutions: real or imaginary.
  • Some contextual situations are better suited to quadratic solutions.
Link to Resource
https://aptv.pbslearningmedia.org/resource/im20-math-ep19-711/what-are-perfect-…

CR Resource Type

Audio/Video

Resource Provider

PBS
Accessibility

Accessibility

Video resources: includes closed captioning or subtitles
License

License Type

CUSTOM

Custom License Type (URL)

https://aptv.pbslearningmedia.org/help/terms-of-use/#.YD-bCNhKiUk

Approved Date

ALSDE LOGO

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